Conformal Maps of

Nonsmooth Surfaces

and Their Applications

Vladimir M. Miklyukov

Exlibris Corporation, Philadelphia2008

Copyright c©2008 by Vladimir M. Miklyukov.

Library of Congress Control Number: 2008903546ISBN: Hardcover 978-1-4363-3693-2Softcover 978-1-4363-3692-5

All rights reserved. No part of this book may be reproduced or transmittedin any form or by any means, electronic or mechanical, including photocopy-ing, recording, or by any information storage and retrieval system, withoutpermission in writing from the copyright owner.This book was printed in the United States of America.

To order additional copies of this book, contact:Xlibris Corporation1-888-795-4274www.Xlibris.comOrders@Xlibris.com

1

Preface

Astonishingly, under the large number of books on conformal maps betweenplane domains, the corresponding theory for surfaces is not represented with aconnected account up to now.

The aim of this book is the introduction to the theory of conformal mapsbetween non-regular surfaces. We are restricted to examination of locally Lips-chitz surfaces. Apparently, such class is minimally necessary in generalizationsand reasonably enough for applications.

Below following problems are touched upon: the existence and uniquenessof maps, the boundary behavior of maps and prime ends of surfaces, theoremsof the Ahlfors and Warschawski type, applications in a problem of the gasdynamics equation and qualitative questions of the theory of minimal surfaces.

Remark that the considered class of problems does not require applica-tions of complex variables. Thus we can courageously offer another title ofthis book: ”Conformal map without complex variables”. Nevertheless, in suchplaces where the application of complex variables is justified by concepts ofconvenient, we use the language of complex variables. The author attemptedto make this presentation maximally simple such that it will be accessible foreverybody including young mathematicians started professional work in thisdomain.

In the book base, there are lectures written by the author for masters of thedepartment of the mathematical analysis and the function theory of VolgogradState University in the 2004/05 academic year. Therefore, as a rule, in thecase of the necessary choice between the account of the result in the maximalgenerality and the showing of a method of its receipt, we preferred the last andrefered to find the generality at original articles.1

We formulate a series of unsolved problems for beginners.The author hopes that the book will be useful to readers, interested in

conformal mappings and their applications.

Vladimir Michaelovich MiklyukovLaboratory ”Superslow processes”,Volgograd State University,University avenue 100, Volgograd 400062, RUSSIAE-mail: miklyuk@mail.ru

1Rephrasing the well-known aphorism: ’We do not sell fish, we sell fishing-tackles’.

Contents

Preface 1

1 Instrumentarium 71.1 Abstract Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Pseudometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 The Distance rΩ as a Finsler Metric . . . . . . . . . . . . . . . . 111.4 The Boundary of Abstract Surfaces . . . . . . . . . . . . . . . . . 131.5 Module of Arc Families . . . . . . . . . . . . . . . . . . . . . . . 131.6 Calculation to the Module . . . . . . . . . . . . . . . . . . . . . . 161.7 Module for Finsler Metrics . . . . . . . . . . . . . . . . . . . . . . 221.8 Condensers on Surfaces . . . . . . . . . . . . . . . . . . . . . . . 241.9 Length and Area Principle . . . . . . . . . . . . . . . . . . . . . . 28

2 Locally Minimal Surfaces 322.1 Surfaces in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 The Laplace-Beltrami Equation . . . . . . . . . . . . . . . . . . . 342.3 The Height Function . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Isothermal Coordinates 423.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Canonical Homeomorphisms . . . . . . . . . . . . . . . . . . . . . 463.3 Nonparametric Surfaces . . . . . . . . . . . . . . . . . . . . . . . 483.4 Proof of Theorem 3.12 . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Proof of Theorem 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . 583.6 Bi-Lipschitz Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 643.7 Quasi-conformal Mappings . . . . . . . . . . . . . . . . . . . . . . 65

4 The Boundary of a Surface 674.1 The Relative Distance . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Prime Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 The Conformal Map T . . . . . . . . . . . . . . . . . . . . . . . . 73

3

4 Contents

4.4 Q∗-homeomorphisms of Surfaces . . . . . . . . . . . . . . . . . . 774.5 Proof of Theorem 4.24 . . . . . . . . . . . . . . . . . . . . . . . . 824.6 Local Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.7 Proof of Theorem 4.30 . . . . . . . . . . . . . . . . . . . . . . . . 91

5 The Speed of Approximation 935.1 Characteristics of a Closeness . . . . . . . . . . . . . . . . . . . . 935.2 Classes BLk and BL . . . . . . . . . . . . . . . . . . . . . . . . . 1015.3 The Complex Dilatation . . . . . . . . . . . . . . . . . . . . . . . 1025.4 The Deviation on Compacts . . . . . . . . . . . . . . . . . . . . . 1065.5 Maps of a disk onto another disk . . . . . . . . . . . . . . . . . . 1135.6 The Measure Stability . . . . . . . . . . . . . . . . . . . . . . . . 1215.7 Proof of Theorem 5.12 . . . . . . . . . . . . . . . . . . . . . . . . 1255.8 Remarks on W 1,2-Majorized Functions . . . . . . . . . . . . . . . 125

5.8.1 The Set P∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.8.2 The Class is not Empty . . . . . . . . . . . . . . . . . . . 127

6 The Area Distortion 1296.1 Graphs over Discs . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 K-Quasi-conformal Maps . . . . . . . . . . . . . . . . . . . . . . 1316.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.4 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 135

7 Ahlfors-Warschawski Theorems 1427.1 Plane Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2 Abutting Subdomains . . . . . . . . . . . . . . . . . . . . . . . . 1447.3 Estimates of Conformal Maps . . . . . . . . . . . . . . . . . . . . 1557.4 Estimates of kΩ(G′), kΩ(G′′) . . . . . . . . . . . . . . . . . . . . 160

8 The Stabilization Speed of Solutions 1658.1 The Gas Dynamics Equation . . . . . . . . . . . . . . . . . . . . 1658.2 The Complex Potential . . . . . . . . . . . . . . . . . . . . . . . 1678.3 Maps onto Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.4 The Uniqueness Problem . . . . . . . . . . . . . . . . . . . . . . . 1718.5 Phragmen-Lindelof Theorems . . . . . . . . . . . . . . . . . . . . 1788.6 Two Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.7 Circular Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.8 Proof of Theorem 8.7 . . . . . . . . . . . . . . . . . . . . . . . . . 1878.9 Half-strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1888.10 Proof of Theorem 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . 192

5

9 Critical Points of a Solution 1949.1 The Nitsche Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1949.2 Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . 1959.3 Maps onto a Half-plane . . . . . . . . . . . . . . . . . . . . . . . 198

9.3.1 Estimates of a Module . . . . . . . . . . . . . . . . . . . . 1999.3.2 Proof of Theorem 9.8 . . . . . . . . . . . . . . . . . . . . 203

9.4 Estimates of the Speed of Growth Solutions . . . . . . . . . . . . 2049.4.1 An Inequality for the Energy Integral . . . . . . . . . . . 2049.4.2 A Conjugate Function . . . . . . . . . . . . . . . . . . . . 2119.4.3 The Growth of a Conjugate Function (I) . . . . . . . . . . 2139.4.4 The Growth of a Conjugate Function (II) . . . . . . . . . 214

9.5 Narrow Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 2169.6 Critical Points of a Solution . . . . . . . . . . . . . . . . . . . . . 221

10 Solutions Close to a Boundary 22610.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22610.2 The Auxiliary Conformal Map . . . . . . . . . . . . . . . . . . . 232

10.2.1 Definitions and Properties . . . . . . . . . . . . . . . . . . 23210.2.2 Conformal Types of Surfaces . . . . . . . . . . . . . . . . 23310.2.3 Some Properties of the Relative Distance . . . . . . . . . 235

10.3 Solution Jumps on a Boundary . . . . . . . . . . . . . . . . . . . 23710.3.1 The Estimate of an Area Graph . . . . . . . . . . . . . . 23710.3.2 Monotonicity of Solutions . . . . . . . . . . . . . . . . . . 23910.3.3 Proof of Theorem 10.17 . . . . . . . . . . . . . . . . . . . 24010.3.4 Points of Quasi-continuity . . . . . . . . . . . . . . . . . . 24110.3.5 Behavior of Solutions at Jump Points . . . . . . . . . . . 24310.3.6 Estimate of a Summary Jump . . . . . . . . . . . . . . . . 246

10.4 The Fatou Type Theorem . . . . . . . . . . . . . . . . . . . . . . 24710.5 Continuity and Quasi-continuity . . . . . . . . . . . . . . . . . . 249

Index 251

References 254

Chapter 10

Solutions Close to aBoundary

We study generalized solutions of minimal surface type equation. We provethat every solution has on the boundary no more than a countable number ofjumps. In particular, every solution, defined in the disc exterior, is continuouslyextendable up to the boundary circle, excepting possibly a countable set ofpoints. For Fatou’s type theorems about angular boundary values, see [109],[83], [115], [100], [120].

10.1 Main Results

Let D ⊂ R2 be a domain and let e ⊂ D be a set of zero linear Hausdorffmeasure.

LetA = (A1(x, ξ), A2(x, ξ)) : (D \ e)× R2 → R2

be a continuous vector function. Suppose that, for every point x = (x1, x2) ∈D \ e and every ξ = (ξ1, ξ2) ∈ R2, the following structure conditions hold:

ν1|ξ|2√

1 + |ξ|2 ≤2∑

i=1

ξi Ai(x, ξ), (10.1)

|A(x, ξ)| ≤ ν2(x) (10.2)

where ν1 is a positive constant and ν2(x) is a positive continuous function.

226

Below we will assume that ν2(x) satisfies

limDn

∫

∂Dn

ν2(x) |dx| = ν∗2 < ∞ (10.3)

where the lower limit is taken over all sequences Dn of subdomains of D withrectifiable boundaries for which Dn ⊂ Dn+1, ∪∞n=1Dn = D. (Here and below,the symbol H means the closure of the set H ⊂ R2 with respect to the topologyR2.)

Consider the equation

2∑

i=1

d

dxiAi(x,∇f) = 0 . (10.4)

As above in Chapter 8, we use the following definition of the generalizedsolution. Denote by Db(f) the subset of D, at every point of which the functionf is not differentiable. By a generalized solution of (10.4), we call a locallyLipschitz function f with the following property. For every bounded subdomain∆, ∆ ⊂ D, with the rectifiable boundary ∂∆, mes1 (∂∆ ∩Db(f)) = 0, and afunction ϕ ∈ Lip∆, the following equality holds

∫

∂∆

ϕ

2∑

i=1

Ai(x,∇f) ni |dx| =∫∫

∆

2∑

i=1

ϕ′xiAi(x,∇f) dx1 dx2 . (10.5)

Here and below, n = (n1(x), n2(x)) is a unit vector of the exterior normal tothe boundary ∂∆.

The set of the discontinuity points of the vector function A has zero linearHausdorff measure, and hence, the contour integral in (10.5) exists.

Exercise 10.6 Prove (or refute !) that classes of generalization solutionsof minimal surface type equations (9.1), (9.2), (9.3), introduced here and inChapter 9, coincide.

2

Let f be a continuous function, defined in the domain D ⊂ R2 with therectifiable boundary. The function f has a finite (or infinite) angular boundaryvalue α at a = (a1, a2) ∈ ∂D if

limx→a

f(x) = α

along every angle C with a vertex at a lying inside D.

227

The following theorem is a version of Fatou’s theorem [147, Chapter I, §5].Every bounded, harmonic in a unit disc B function f has angular boundaryvalues a.e. on the circle ∂B.

Moreover, there are examples of unbounded harmonic functions which haveno angular boundary values on sets H ⊂ ∂B with linear measure mes1H > 0[28, Chapter 2, §10].

Problem. (J.C.C. Nitsche [137, Chapter VII, n. 4]) Do valid Fatou’s typetheorems exist for solutions of (2.20)?

The following result was obtained in [115].

(A) Every solution of the minimal surface equation (2.20) has finite or infi-nite angular boundary values a.e. on ∂B.

Let’s emphasize that here we do not suppose additional restrictions for so-lutions.

At the same time, it is necessary to remark that the behavior of solutionsof (2.20) depends on the specific structure of the domain where these solutionsare defined. Namely, for the solutions, defined in the exterior of a disc B, thefollowing statement holds [115].

(B) Every solution of the minimal surface equation (2.20), defined over R2 \B, is continuously extended a.e. to the boundary, i.e. a finite limit

limx→a

f(x) = α, x ∈ R2 \B

exists a.e. on the circle ∂B.

Theorem 10.7 Let D ⊂ R2 be a domain with the Jordan rectifiable boundary.Every generalized solution of the equation (10.4) with the structure conditions(10.1), (10.2), (10.3) has finite or infinite angular boundary values a.e. on ∂D.

On generalizations, see [100], [120].

Example 10.8 A solution of (2.20) can have infinite values on a set of thepositive linear measure. Consider the Scherk surface

f(x1, x2) = logcos x2

cos x1,

defined over the square

Q =

(x1, x2) : −π

2< xi <

π

2(i = 1, 2)

.

This surface is minimal, and the function f equals±∞ on horizontal and verticalsides of the square boundary. In the vertices of the square, the function f |∂Q

has jumps.

228

2

Let D ⊂ R2 be a simply connected domain with the Jordan boundary ∂Dand let O ∈ D be a fixed point. Let U ⊂ D be an open set. We denote:[U ] = U \ ∂D, ∂′U = [U ] \ U.

Definition 10.9 Let f be a continuous function defined in D. We will calla ∈ ∂D by the point of quasi-continuity f if there is a sequence of subdomainsDk∞k=1 of the domain D with properties:

(α) every ∂′Dk separates a from the fixed point O;

(β) ∩k [Dk] = ∅; length ∂′Dk → 0 for k →∞;

(γ) lim infk→∞ osc(∂′Dk, f) = 0 .We will call all remaining points of ∂D by jump points of the function f.

Define the following nonlocal characteristic of a boundary point a ∈ ∂D.We put

δ(a, f,O) = infγ

maxosc (γ, f), length (γ)

where the infimum is taken over all arcs γ ⊂ D, γ ∩ ∂D 6= ∅, separating pointsa and O in D.

229

It is clear that f is quasi-continuous at a point a ∈ ∂D if and only ifδ(a, f, O) = 0.

On the structure of solutions of (2.20) close to jump points, see Lancaster[83]. We prove that boundary quasi-continuity points of solutions f of theminimal surface type equation are typical.

Consider the set H of all piecewise continuous functions h : R → R withproperties:

(i) 0 ≤ h(t) ≤ 1 for all t ∈ R;

(ii)+∞∫−∞

h(t) dt ≤ h0 < ∞, h0 = const.

Theorem 10.10 Let D ⊂ R2 be a simply connected domain bounded with asimple rectifiable Jordan curve ∂D, O ∈ D. Let f : D → R be an arbitrarysolution of an equation (10.4) with structure restrictions (10.1), (10.2), (10.3).

Then for every h ∈ H, the function

w(x) =

f(x)∫

−∞h(t) dt

is quasi-continuous at all points a ∈ ∂D, except only in a countable set.Moreover, if a1, a2, . . . are jump points of f , then

3∑i=1

max

π − K2

πδ2(ai, w,O), 0

+

+2∞∑

i=4

arcsin(

12 exp

− K2

δ2(ai, w, O)

)≤ π

(10.11)

where

K = 2√

π

(h0

ν∗2ν1

+ 2mes2 (D))1/2

.

Example 10.12 Suppose that |f | < 1 everywhere in D. We put

h(t) =

1 for t ∈ [−1, 1] ,0 for |t| > 1 .

For the auxiliary function

w(x) =

f(x)∫

−∞h(t) dt ,

230

we have 0 ≤ w(x) ≤ 2. Points of quasi-continuity of f(x) are points of quasi-continuity of w(x). Moreover,

δ(ak, w, O) = δ(ak, f, O) for all k = 1, 2, . . . .

2

Example 10.13 In the case of an unbounded function f(x), we can put

h(t) =1

1 + t2.

Then

w(x) =

f(x)∫

−∞

dt

1 + t2= arctg f(x) +

π

2.

Here Theorem 10.10 guarantees that arctg f(x) is quasi-continuous. It isclear that in the general case, the quasi-continuity of this function does notimply the quasi-continuity of f(x). The function f(x) from Example 10.8 isnot quasi-continuous in every boundary point. However, arctg f(x) is quasi-continuous everywhere on the boundary of the square except its vertices.

2

As a corollary of Theorem 10.10, we have the following statement.

Theorem 10.14 Let E = x = (x1, x2) ∈ R2 : x21 + x2

2 > 1 be an exterior ofa unit circle C. Let D ⊂ E be a domain and Γ ⊂ (C ∩ ∂D) be a boundary arc.

Then every solution f of the minimal surface equation (2.20) in D is con-tinuously extendable at an arbitrary point a ∈ Γ, except at most, in a countableset.

Open questions 10.15 1) Find analogs of Theorem 10.14 for solutions ofminimal surface type equations. 2) Prove (or refute) Theorem 10.14 with theexchange in its formulation the boundary circle replaced by a concave curve ofa sufficient general form.

231

10.2 The Auxiliary Conformal Map

10.2.1 Definitions and Properties

Let Ω be a graph of a locally Lipschitz function x3 = f(x1, x2), defined over adomain D ⊂ R2, and let

ds2Ω = (1 + f ′2x1

) dx21 + 2f ′x1

f ′x2dx1 dx2 + (1 + f ′2x2

) dx22 (10.16)

be the square of its line element on Ω.Fix arbitrarily a point a ∈ D where f is differentiable. The function f ∈

LipD, and hence, by Rademacher’s theorem, it is differentiable a.e. in D.The quadratic form ds2

Ω is positive determined. Thus, an infinitesimal circlewith respect to ds2

Ω and a center at a is an infinitesimal ellipse with respect tothe Euclidean metric. Let θ(a), 0 ≤ θ < π, and p(a), p ≥ 1, be characteristicsof this ellipse (i.e. the angle between its largest axis and the Ox1-axis, and theratio of the largest axis to the smaller, respectively).

It is not difficult to see that

θ(a) =π

2+ arctg

f ′x2(a)

f ′x1(a)

, p(a) =√

1 + |∇f(a)|2.

We consider a quasi-conformal map u = u(x) of the domain D into theu = (u1, u2)-plane with characteristics which coincide a.e. in D with θ(x) andp(x). Since ess supD′ p(x) < ∞ for every bounded subdomain D′, D′ ⊂ D, thensuch map exists and defined up to conformal maps in the u−plane (see Theorem3.49).

We assume that the following properties of quasi-conformal maps u : D →R2 are known.

(1) A map u = u(x) is differentiable a.e. in D, and at every point of the dif-ferentiability it transforms infinitesimal ellipses with characteristics θ(x), p(x)onto infinitesimal circles (see [17, §1]).

(2) A map u = u(x) has the first generalized Sobolev derivatives, squareintegrable locally in D, i.e. of W 1,2

loc (D) (see [17, §4]).

(3) Denote by G = u(D) the image of D under u = u(x), and by x =x(u) = (x1(u), x2(u)) - the inverse map. The characteristic p(u) of the inversemap x : G → D is locally bounded in the domain G (see [17, §1]), and by theproperty (2) the map x = x(u) belongs to W 1,2

loc (G) also.

We put x3(u) = f(x(u)). Since f is locally Lipschitz in D, then by theproperty (3) the function x3(u) ∈ W 1,2

loc (G).

232

From (2) it follows that the vector function

χ(u) = (x1(u), x2(u), x3(u))

also belongs to W 1,2loc (G). It is clear that χ(u) is differentiable a.e.

The vector function χ : G → Ω realizes a bijective map of G = u(D) ontoΩ. Let a ∈ G be a point where χ is differentiable. Denote by j : F → D theprojection of Ω onto D. The map χ : G → Ω is a composition of maps j−1 andx(u). Thus χ : G → Ω is conformal a.e. and a.e. on G it has properties (??).Variables u1, u2 are isothermal coordinates on Ω.

10.2.2 Conformal Types of Surfaces

By the stated above, we say that a surface Ω is of parabolic conformal type ifG = R2 and of hyperbolic conformal type if G 6= R2.

Fix a function h ∈ H. Denote by W the graph of the function

w(x) =

f(x)∫

−∞h(t) dt.

Theorem 10.17 Let D ⊂ R2 be a simply connected domain bounded by asimple Jordan rectifiable curve. If f is a generalized solution of an equation(10.4) in D with structure restrictions (10.1), (10.2), (10.3), then for everyh ∈ H, the graph W of x3 = w(x) is of hyperbolic conformal type.

Some indications of parabolicity and hyperbolicity of the conformal type ofa function graph can be obtained from Theorem 1.36. Below we formulate twosuch indications.

Theorem 10.18 Let f be a locally Lipschitz function and let Ω be its graph.If Ω is defined over a disc |x| < R, 0 < R ≤ ∞, and for a number 0 < r < R

it is fulfilled ∫∫

r<|x|<R

1 + 〈∇f, x⊥〉2√1 + |∇f |2

dx1dx2

|x|2 < ∞ , (10.19)

then Ω is of hyperbolic conformal type.If Ω is defined over the plane R2 and

limR→∞

1R2

∫∫

1<|x|<R

1 + 〈∇f, (x/|x|)〉2√1 + |∇f |2 dx1dx2 = 0 , (10.20)

233

then Ω is of parabolic conformal type.Here by x⊥ we denote a unit vector in R2, orthogonal to the vector x and

forming the angle 3π/2 in the direction from x⊥ to x.

The proof is not difficult. Fix 0 < r < R < ∞. Let Γ(r,R) be the familyof all locally rectifiable arcs γ, joining boundary circles and lying in the ringr < |x| < R. Because modΩ Γ(r,R) is the conformal invariant, then Ω is ofparabolic type if and only if modΩ Γ(r,R) = 0. By the remark of Section 1.7,we have

R∫

r

dτ

τ

2π∫

0

1 + 〈∇f, (x/|x|)〉2√1 + |∇f |2 dθ

−1

≥

≥ modΩΓ(r,R) ≥2π∫

0

dθR∫

r

1 + 〈∇f, x⊥〉2τ

√1 + |∇f |2 dτ

.

By the Cauchy integral inequality, we obtain

(R− r)2 ≤R∫

r

dτ

τ

2π∫

0

1 + 〈∇f, (x/|x|)〉2√1 + |∇f |2 dθ

×

×R∫

r

τ dτ

2π∫

0

1 + 〈∇f, (x/|x|)〉2√1 + |∇f |2 dθ .

Thus

(R− r)2∫∫

r<|x|<R

1 + 〈∇f, (x/|x|)〉2√1 + |∇f |2 dx1dx2

≤R∫

r

dτ

τ

2π∫

0

1 + 〈∇f, (x/|x|)〉2√1 + |∇f |2 dθ

that proves (10.20).

234

For the proof of (10.18), we observe that from Cauchy’s inequality it follows

2π∫

0

dθ

2

≤2π∫

0

dθ

τ

R∫

r

1 + 〈∇f, x⊥〉2√1 + |∇f |2 dτ

2π∫

0

τ dθ

R∫

r

1 + 〈∇f, x⊥〉2√1 + |∇f |2 dτ

and

4π2

2π∫

0

dθ

R∫

r

1 + 〈∇f, x⊥〉2τ

√1 + |∇f |2 dτ

≤2π∫

0

τ dθR∫

r

1 + 〈∇f, x⊥〉2√1 + |∇f |2 dτ

.

Thus we deduce

4π2

∫∫

r<|x|<R

1 + 〈∇f, x⊥〉2√1 + |∇f |2

dx1dx2

|x|2≤

2π∫

0

τ dθR∫

r

1 + 〈∇f, x⊥〉2√1 + |∇f |2 dτ

.

The condition (10.19) implies that modΩΓ(r,R) > 0 and Ω is of parabolicconformal type. 2

On indications of conformal type, see also [162], [126], [110], [51], [167] etc.

10.2.3 Some Properties of the Relative Distance

Let Ω ⊂ R3 be a graph of a locally Lipschitz function f defined over a simplyconnected domain D ⊂ R2, and let O′ ∈ Ω be a fixed point.

For a pair of points p, q ∈ Ω, let ρ(p, q; O′,Ω) be the relative distance,introduced in Chapter 4. By ∂Ω, as above, we denote the boundary of Ω withrespect to the metric ρ, i.e. it will be the set of all sequences pn of points in Ω,fundamental with respect to the relative metric ρ and having no accumulationpoints in Ω.

If Ω is the graph of the function x3 ≡ const over the domain D ⊂ R2,then we identify Ω with D, and the relative boundary ∂Ω coincides with theCaratheodory boundary of D.

Suppose that D is simply connected and its image G = u(D) is differentfrom the entire plane R2. Because relations (2.3) are invariant under conformaltransforms in the u = (u1, u2)-plane, then without loss of generality, we canassume that the domain G is bounded.

235

Consider the above described conformal map

χ(u) = (x1(u), x2(u), x3(u)) : G → Ω, u = (u1, u2) . (10.21)

We put O′′ = χ−1(O′) and denote by r(G) the Euclidean distance from O′′

to ∂G.The following statement is a special case of Theorem 4.6.

Theorem 10.22 If the area mes2 (Ω) < ∞, then for an arbitrary pair ofpoints p, q ∈ G, satisfying the condition

ρ(p, q; O′′, G) < min1,116

r4(G), (10.23)

it is fulfilled

ρ(χ(p), χ(q); O′, Ω) ≤ K log−1/2 1ρ(p, q; O′′, G)

. (10.24)

HereK = 2

√π mes2 (Ω).

For the proof, it is sufficient to remark that the surface Ω, given by thegraph of a locally Lipschitz function f , is locally bi-Lipschitz. It is obvious,since for every subdomain D′ ⊂⊂ D, the relation (10.16) implies

|dx|2 ≤ ds2Ω = (1 + f ′2x1

) dx21 + 2f ′x1

f ′x2dx1 dx2 + (1 + f ′2x2

) dx22 ≤ C(f, D′) |dx|2

where C(f, D′) is a constant. 2

The estimate (10.24) implies that every fundamental (with respect to therelative distance ρ(p, q; O′′, G)) sequence an ⊂ G turns to a fundamental(with respect to the metric ρ(χ(p), χ(q); O′, Ω)) sequence χ(an) ⊂ Ω. Thuswe obtain the following statement.

Corollary 10.25 Under conditions described in Theorem 10.14, a conformalmapping (10.21) is continuously extended up to a continuous mapping of therelative boundary ∂G onto the relative boundary ∂Ω.

It should be noted that the following important property of the projectionj : F → D.

236

Lemma 10.26 Let Ω be the graph of a locally Lipschitz function, defined overa simply connected domain D ⊂ R2, and j(O′) = O.

Then for every pair of points p, q ∈ Ω, it fulfills

ρ(j(p), j(q); O, D) ≤ ρ(p, q; O′, Ω) . (10.27)

Proof. It is sufficient to remark that the projection j : F → D does notincrease lengths of curves. 2

Let x(u) = (x1(u), x2(u)) : G → D be a map realized by components x1(u),x2(u) of the vector function (10.21).

Corollary 10.28 If mes2 (Ω) < ∞, then for every pair of points p, q ∈ Gwith the property (10.23), it fulfills

ρ(x(p), x(q); O, D) ≤ K log−1/2 1ρ(p, q;O′′, G)

where K is the constant of Theorem 10.22.

The proof follows from Theorem 10.22 and Lemma 10.26. 2

10.3 Solution Jumps on a Boundary

We prove Theorem 10.10.At first we observe that the statement is trivial if f ≡ const. Thus we can

assume that f 6≡const.

10.3.1 The Estimate of an Area Graph

Let f be a locally Lipschitz solution of the equation (10.4) with structure con-ditions (10.1)-(10.3). We assume that the domain D is simply connected andbounded with a simple Jordan rectifiable curve.

Lemma 10.29 Under described suppositions, the following inequality holds

mes2 (W ) ≤ h0ν + 2mes2 (D). (10.30)

237

Proof. Fix a sequence of domains Dk, k = 1, 2, . . . , such that

Dk ⊂ Dk+1, ∪∞k=1Dk = D.

The function w(x) belongs to the class Lip (Dk), k = 1, 2, . . .. Choosingϕ = w(x) in (10.5), we have

∫∫

Dk

2∑

i=1

f ′xiAi(x,∇f) h(f(x)) dx1 dx2 =

∫

∂Dk

w(x)2∑

i=1

Ai(x,∇f)ni |dx|

≤∫

∂Dk

w(x) |A(x,∇f)| |dx|.

Since 0 < w(x) ≤ h0 < ∞, then

∫∫

Dk

2∑

i=1

f ′xiAi(x,∇f)h(f(x)) dx1 dx2 ≤ h0

∫

∂Dk

|A(x,∇f)| |dx|.

Passing to the limit as k →∞ and using the structure conditions (10.1)-(10.3),we find

ν1

∫∫

D

|∇f |2√1 + |∇f |2 h(f(x)) dx1 dx2 ≤ h0 ν∗2 . (10.31)

From (10.31) we deduce∫∫

D

√1 + |∇f |2h(f(x)) dx1 dx2 ≤ h0

ν∗2ν1

+∫∫

D

h(f(x))√1 + |∇f |2 dx1 dx2.

From here, remembering that 0 ≤ h(f(x)) ≤ 1, we arrive to the estimate∫∫

D

|∇w(x)| dx1 dx2 =∫∫

D

|∇f(x)|h(f(x)) dx1 dx2 ≤ h0ν∗2ν1

+ area (D).

Because √1 + |∇w(x)|2 < 1 + |∇w(x)|,

we find finally∫∫

D

√1 + |∇w|2 dx1 dx2 ≤ h0

ν∗2ν1

+ 2 area (D)

that is equivalent to (10.30). 2

238

10.3.2 Monotonicity of Solutions

The square of the length element on the surface W is given by the formula

ds2W =

2∑

i,j=1

gij(x) dxi dxj , gij(x) = δij + w′xiw′xj

dxi dxj (i, j = 1, 2)

where δij (i, j = 1, 2) is the Kronecker symbol.Let (gij(x)) = (gij)−1(x) be the matrix inverse to (gij). Simple calculations

imply

gij = δij −w′xi

w′xj

1 + |∇w|2 .

For an arbitrary ξ ∈ R2, we put

|ξ|2W =2∑

i,j=1

gij(x) ξiξj .

It is easy to check that

|∇w|2W =1 + w′2x2

1 + |∇w|2 w′2x1− 2

w′x1w′x2

1 + |∇w|2 w′x1w′x2

+

+1 + w′2x1

1 + |∇w|2 w′2x2=

|∇w|21 + |∇w|2 .

Since ∇w(x) = h(f(x))∇f(x), the relation (10.31) implies∫∫

D

|∇w|2√h2(f) + |∇w|2 dx1 dx2 =

∫∫

D

|∇f |2√1 + |∇f |2 h(f(x)) dx1 dx2 ≤ h0

ν∗2ν1

.

From here, taking into account that h(f(x)) ≤ 1, we arrive to the inequality∫∫

D

|∇w|2√1 + |∇w|2 dx1 dx2 ≤ h0

ν∗2ν1

,

or ∫∫

D

|∇w|2W√

1 + |∇w|2 dx1 dx2 ≤ h0 ν. (10.32)

239

Lemma 10.33 Every generalized solution f of the equation (10.4) with struc-ture conditions (10.1), (10.2), (10.3) is monotone in the Lebes- gue sense.

Proof. Indeed, for example, suppose that a ∈ D is a point of the stronglocal maximum. For sufficiently small ε > 0, the set x ∈ D : f(x) > f(a)− εcontains a precompact component U ⊂ D, a ∈ U .

We have f(x)− f(a) + ε = 0 on the boundary of U . For a.e. ε > 0, curves∂U are rectifiable. Choosing in (10.5) the function φ = f(x)− f(a) + ε, for a.e.ε > 0 we can write

∫∫

U

2∑

i=1

f ′xiAi(x,∇f) dx1dx2 =

∫

∂U

(f(x)− f(a) + ε)2∑

i=1

Ai(x,∇f)ni |dx| = 0.

Thus from (10.1), it follows∫∫

U

|∇f |2√1 + |∇f |2 dx1dx2 = 0,

i.e. ∇f ≡ 0 and f ≡ const in U. We have the contradiction with the assumptionthat a is the point of the strong local maximum. 2

10.3.3 Proof of Theorem 10.17

As above, with an auxiliary quasi-conformal map u = u(x) : D → R2, weintroduce isothermal coordinates (u1, u2) on the surface W .

Consider the mapping x = x(u) = (x1(u), x2(u)), inverse to u = u(x), andthe function x3(u) = w(x(u)). The vector function

χ(u) = (x1(u), x2(u), x3(u))

realizes the one-to-one conformal mapping of G = u(D) onto W .Complete the proof of the theorem. At first we remark that by Lemma

10.33 the function w(x) is monotone in D, and since the map x(u) : D → G ishomeomorphic, then w∗(u) = w(x(u)) is also monotone in G. Because x(u) isquasi-conformal and f is locally Lipschitz, then w∗ ∈ W 1,2

loc (G).Suppose that the function w∗ is defined over the whole of plane R2. Fix

R > 0 and t > 1. Let B(τ) and S(τ) be the disc and the circle with the centerat u = 0 and the radius τ , respectively.

240

As the proved above inequality (4.14), we prove that

infR≤τ≤tR

osc(S(τ), w∗) ≤(

2π I(R)log t

)1/2

(10.34)

whereI(R) =

∫∫

|u|<tR

|∇w∗|2du1 du2.

The function w∗ is monotone, and hence, for every t > 1, it fulfills

osc(B(R), w∗) ≤ osc(S(R), w∗) ≤ infR≤τ≤tR

osc(S(τ), w∗).

Consequently,

osc(B(R), w∗) ≤(

2π I(∞)log t

)1/2

.

Taking into account (10.32), we find

I(∞) =∫∫

R2

|∇w∗|2 du1 du2 =∫∫

D

|∇w|2W√

1 + |∇w|2 dx1 dx2 ≤ h0 ν,

and hence,

osc(B(R), w∗) ≤(

2π h0 ν

log t

)1/2

.

Setting now t → ∞, we conclude that w∗ ≡ const on B(R). But R > 0 isarbitrary, and hence, w∗ ≡ const in the whole of plane R2. From here, it followsthat w ≡ const in D.

However, if w ≡ const in D, then the map u = u(x) : D → R2 is conformalwith respect to the Euclidean metric, and the image of D can not be the wholeof plane R2. We have a contradiction. 2

10.3.4 Points of Quasi-continuity

Because the simply connected domain G 6= R2 and the mapping u(x) : D → Gis defined up to conformal mappings in the u−plane, we are right to assumethat G is a unit disc with the center at O′′.

We will show that all points of ∂G are points of the quasi-continuity of w∗.Fix a point a ∈ ∂G. Let S(a, τ) be a component of the set

u ∈ G : |u− a| = τseparating the point a ∈ ∂G from the origin O′′, 0 < τ < 1.

241

As in (10.34), we prove that for every R ∈ (0, 1) and every t ∈ (0, 1), thereexists τ0 ∈ (tR, R) such that

inftR<τ<R

osc (S(a, τ), w∗) ≤(

2π I

log 1t

)1/2

whereI =

∫∫

G

|∇w∗|2 du1 du2 .

Choosing now tn → 0, we find a sequence τn → 0 along which

osc(S(a, τn), w∗) → 0 . (10.35)

This means that a is the quasi-continuity point of w∗ at a ∈ ∂G.

By Lemma 10.29 the integral∫∫

D

√1 + |∇w|2 dx1 dx2 < ∞.

Using Theorem 10.14 and Lemma 10.26, we conclude that the mapping

x(u) = j χ(u) : G → D

is continuous up to the boundary of the unit disc ∂G.Denote by x(u) : G → D, x(u)|G = x(u), the map between closed domains

obtained with the continuous extension of x(u) : G → D to ∂G.Moreover, by Corollary 10.28 for arbitrary points

p, q ∈ G with ρ(p, q;O′′, G) < 1/16 ,

we haveρ(x(p), x(q); O,D) ≤ K1 log−1/2 1

ρ(p, q; O′′, G)where

K1 = 2√

π mes2 (W )

is the constant.By Lemma 10.29

K1 ≤ 2√

π (ν h0 + 2 area (D))1/2 = K .

In particular, if points p and q lie on the boundary circle and |p − q| < 1/16,then ρ(p, q; O′′, G) = |p− q|. Thus the found estimate takes the form

ρ(x(p), x(q); O, D) ≤ K log−1/2 1|p− q| |p− q| < 1

16. (10.36)

242

Since the mapping x(u) : G → D is continuous, then the preimage x−1(a)of a point a ∈ ∂D is a closed set on the unit circle ∂G. Moreover, for every pairof points a 6= b on ∂D, it is fulfilled

x−1(a) ∩ x−1(b) = ∅ .

The set x−1(a) is connected. Indeed, because ∂D is the simple Jordan curve,for sufficiently small ε > 0 the set Uε = x ∈ D : |x− a| < ε is connected. Themapping x(u) : G → D is homeomorphic, and hence, preimages x−1(Uε) areconnected for all ε ∈ (0, ε0) where ε0 > 0 is a sufficiently small number. Thusthe set

x−1(a) = ∩ε>0U ε

is also connected.So, for an arbitrary point a ∈ ∂D, the set x−1(a) can be either an isolated

point or a closed connected arc.Since the amount of non-overlapping arcs on a circle is no more than count-

able, then the inverse mapping x−1(x) is defined everywhere on ∂D, except, atmost, in a countable set E ⊂ ∂G.

We show that every point a ∈ ∂D \ E is the quasi-continuity point of thefunction w(x).

Let b = x−1(a) be the preimage of the point a ∈ ∂D \ E. By (10.35) thereexists a sequence of subdomains Gn, ∂′Gn = S(b, τn) with properties:

every arc ∂′Gn separates b from the origin O;

length (∂′Gn) → 0 and osc (∂′Gn, w∗) → 0 as n →∞;

finally, ∩∞n=1[Gn] = ∅.

We put Dn = x(Gn), n = 1, 2, . . .. Because the map x(u) : G → D ishomeomorphic, every arc ∂Dn = x(S(b, τn)) separates a ∈ ∂D from O.

The map x(u) : G → D is continuous, and hence, length ∂′Dn → 0 and∩∞n=1[Dn] = ∅.

Finally, osc (∂′Dn, w) = osc(S(b, τn), w∗), and hence, osc (∂′Dn, w) → 0 asn →∞.

Thus the point a ∈ ∂D \E has all properties, necessary to define the pointsof quasi-continuity of w(x).

10.3.5 Behavior of Solutions at Jump Points

Fix a jump point a ∈ E. Its preimage x−1(a) is a subarc β of the circle |u| = 1.We estimate its length l(β). Let ξ ∈ β be the middle of the arc. It is easy now

243

to calculate

r = 2 sinl(β)4

,

which would be the distance from ξ to arc ends.Suppose that r < 1, i.e. l(β) < 2

3π. Consider the family of circles S(ξ, τ)with the center at ξ and radii τ, r < τ < 1. We put Cτ = G ∩ S(ξ, τ).

By (4.12) we have

1∫

r

l2(χ(Cτ ))τ

dτ ≤ 2π

∫∫

∆r,1

3∑

i=1

|∇xi(u)|2 du1 du2.

Remark now that for every τ ∈ [r, 1], it is fulfilled

l(χ(Cτ )) ≥ maxosc (x(Cτ ), w), length (x(Cτ )) ≥ δ(a,w, O),

and hence,

δ2(a, w, O) ≤ 4π log−1 1r

mes2 (W ).

Thus, taking into account the estimate (10.30), we arrive to the inequality

δ2(a,w, O) ≤ K2

log 1/r

where K is the constant of Theorem 10.10.From here,

r = 2 sinl(β)4

≥ exp− K2

δ2(a,w, O),

and we have the following estimate of l(β) for ’small’ δ(a,w,O):

l(β) ≥ 4 arcsin(

12

exp− K2

δ2(a, w, O))

for l(β) <23π. (10.37)

Find the estimate of l(β) for ’big’ δ(a, w, O).Let a ∈ ∂D ∩ E be a jump point in which 2

3π ≤ l(β) ≤ 2π. Without loss ofgenerality, we can assume that the arc β = x−1(a) is described by relations

β = u = (u1, u2) : u21 + u2

2 = 1, − l(β)2

≤ arctgu2

u1≤ l(β)

2.

Fix the segment

γ = u = (u1, u2) : 0 ≤ u1 ≤ 1, u2 = 0

244

and denote by p1 and p2 the boundary points of the unit disc with the cut γ,lying in the intersection of the circle |u| = 1 and the upper and lower edges ofγ respectively.

Let T : G → V be the conformal mapping of the disc |u| < 1 with thecut γ onto the half-disc V = v = (v1, v2) : v2

1 + v22 < 1, v2 > 0 for which

T (p1) = (1, 0), T (p2) = (−1, 0) and T (0, 0) = (0, 0).Under the map T , the arc β ⊂ ∂G is corresponding to

η =

v = (v1, v2) :l(β)4

≤ arctgv2

v1≤ π − l(β)

4

.

For every 0 < k < cos l(β)4 , the rectilinear segment

ζ(k) = v = (v1, v2) : −√

1− k2 < v1 <√

1− k2, v2 = k

separates in the domain V the arc η from the point (0, 0). Its image ζ∗(k) =x T−1(ζ(k)) is an arc, separating points a and O in D. Thus for every k ∈(0, cos l(β)

4 ) it is fulfilled

l2(ζ∗(k)) ≤

∫

ζ(k)

∣∣∣∣∂χ∗

∂v1

∣∣∣∣ dv1

2

≤ 2√

1− k2

∫

ζ(k)

∣∣∣∣∂χ∗

∂v1

∣∣∣∣2

dv1

where χ∗ = χ T−1.Further,

cos l(β)/4∫

0

l2(ζ∗(k))√1− k2

dk ≤ 2∫∫

V

∣∣∣∣∂χ∗

∂v1

∣∣∣∣2

dv1 dv2 ≤ 2 mes2 (W ).

Taking into account that

l(ζ∗(k)) ≥ δ(a, w, O) for all k ∈ (0, cosl(β)4

) .

From here we obtain

δ2(a,w,O)

cos l(β)/4∫

0

1√1− k2

dk ≤ 2mes2 (W ) ,

or

δ2(a,w,O)(

π

2− l(β)

4

)≤ K2

2π

245

where K is the constant of Theorem 10.10.Thus we arrive to the estimate

l(β) ≥ max

2π − 2K2

π δ2(a,w, O), 0

. (10.38)

10.3.6 Estimate of a Summary Jump

Let ak be jump points of the function w(x) and let βk be corresponding closedarcs on |u| = 1, k = 1, 2, . . .. We are right to assume that

l(β1) ≥ l(β2) ≥ . . . ≥ l(βk) ≥ . . . .

Since βi ∩ βj = ∅ for i 6= j, then

∞∑

k=1

l(βk) ≤ 2π.

Beginning at least with k = 4, it is fulfilled l(βk) < 23π, and we can use the

estimate (10.37). We have

4∞∑

k=4

arcsin(

12

exp− K2

δ2(a,w, O))≤

∞∑

k=4

l(βk). (10.39)

For every point a1, a2, a3, we use the estimate (10.38). Then

3∑

k=1

max2π − 2K2

πδ2(a,w, O), 0 ≤

3∑

k=1

l(βk). (10.40)

Combining (10.39) and (10.40), we arrive to the inequality

3∑

k=1

max2π − 2K2

π δ2(a,w,O), 0+ 4

∞∑

k=4

arcsin(

12

exp− K2

δ2(a, w, O))

≤3∑

k=1

l(βk) +∞∑

k=4

l(βk) ≤ 2π.

Thus Theorem 10.10 is proved completely. 2

246

10.4 The Fatou Type Theorem

Prove Theorem 10.7. Let f : D → R be a function, monotone in the Lebesguesense, and let a ∈ ∂D be a point. We call that f has the Lindelof property ata if the existence finite limits of f along ways Γ1 and Γ2, leading to this point,implies the existence of a limit of f along every way, leading to a and lyingbetween Γ1 and Γ2.

The proof of Theorem 10.7 is based on the following statement.

Lemma 10.41 If a function is monotone in the Lebesgue sense and quasi-continuous at a boundary point, then this function has the Lindelof property atthis point.

For the proof we fix a point of quasi-continuity a ∈ ∂D and a sequence ofsubdomains Dk∞k=1 of D with properties (α), (β), (γ) of the definition 10.9.

Let Γ1 and Γ2 be arbitrary ways, leading to a, along which f has limitsα1 and α2 respectively. Denote by Uk the subdomain of Dk lying betweenΓ1, Γ2, ∂Dk and ∂Dk+1. Without loss of generality, we can assume that suchsubdomain is defined for every k = 1, 2, . . ..

For k = 1, 2, . . ., we put

γ1k = ∂Uk ∩ Γ1, γ2k = ∂Uk ∩ Γ2, γ3k = ∂Uk ∩ ∂Dk, γ4k = ∂Uk ∩ ∂Dk+1 .

The function f has finite limits along Γ1 and Γ2. Therefore,

osc (γik, f) → 0 as k →∞ for i = 1, 2.

Moreover, by (γ) of the definition 10.9, we have

osc (γik, f) → 0 as k →∞ for i = 3, 4.

Thus, taking into account that

∂Uk = ∪4i=1γik,

we can concludeosc (∂Uk, f) → 0 as k →∞.

However, f is monotone in the Lebesgue sense, and therefore,

osc (Uk, f) → 0 as k →∞.

This means that f has a limit along the sequence Uk and, in particular,α1 = α2. Thus f has the Lindelof property at a. 2

247

Proof of Theorem 10.7. Let f be a solution of (10.4) with structure restric-tions (10.1), (10.2), and (10.3) in D. We put Φ(x) = arctg f(x).

Choose h(t), as in Example 10.13. By (10.30), we have∫∫

D

|∇Φ(x)| dx1 dx2 < ∞.

Denote by Dc the intersection of D and the line (x1, x2) ∈ R2 : x1 = c,and by Γc - the intersection of ∂D and the same line. Using Fubini theorem,we conclude that for a.e. c, belonging to the projection of D onto the axis 0x1,the following relation holds

∫

Dc

|∇Φ(c, x2)| dx2 < ∞.

From here for every pair of points (c, x′2), (c, x′′2) which belong to one and thesame connected component of Dc, we obtain

|Φ(c, x′′2)− Φ(c, x′2)| ≤x′′2∫

x′2

|∇Φ(c, x2)| dx2 (x′2 < x′′2).

Thus, Φ(c, x2) is uniformly continuous on every component of the set Dc, andhence, at every point a ∈ Γc, it has a limit along Dc.

We turn the coordinate system in the (x1, x2)−plane in the angle α ∈ Fwhere F is a countable set dense everywhere on (0, 2π), and reason every timeas above. Thus we can touch a.e. point a ∈ ∂D with a corner of any angle,close to π and such that Φ has limits along every side of this corner.

Suppose that a ∈ ∂D be a point of the described form. By Theorem 10.10the function Φ = arctg f is quasi-continuous everywhere on ∂D except, possibly,in a countable set, and without loss of generality, we are right to assume thata is a point of the quasi-continuity of f . By Lemma 10.33 the function f (and,hence, Φ = arctg f) is monotone in the Lebesgue sense. Thus by Lemma 10.41,this function has Lindelof’s property at a.

Choose arbitrarily a corner along sides of which Φ has limits. These limitsequal a number α, |α| ≤ π

2 , and the constant α is also the limit of Φ along thecorner domain. But magnitude of this corner can be chosen arbitrarily close toπ, and hence, Φ has the angular boundary value α.

From here, f = tg Φ has the finite or infinite angular boundary value tg αat this point. 2

248

10.5 Continuity and Quasi-continuity

Below we prove Theorem 10.22. We will need a special case of Finn’s lemma[38]. In the form, necessary below, this statement is contained in [141, Lemma10.2].

Lemma 10.42 Let ∆ be a subdomain, lying in a ring 1 < |x| < b, and let γbe a set of boundary points of ∆ which are not lying on the unit circle. Let f bea solution of the minimal surface equation (2.20) in ∆ such that, for all pointsx ∈ γ

m ≤ f(x) ≤ M

where m and M are constants.Then, everywhere in ∆, the following inequality holds

m− arccosh b ≤ f(x) ≤ M + arccosh b. (10.43)

Proof of Theorem 10.14. The function f can not be identically to ±∞ onan arc Γ1 ⊂ Γ (see [141, §10]). Thus there exists an everywhere dense set onΓ, such that in its every point, it is possible to touch outwards D with an arcalong which f is bounded. Using Lemma 10.42, from here we conclude thatin arbitrary strongly inner subarc Γ1 ⊂ Γ, limiting values of f are bounded.Thus it is sufficient to prove the statement in case if D is simply connected andbounded by a simple Jordan rectifiable curve and f is bounded in D.

By Theorem 10.10 the function f is quasi-continuous everywhere on ∂D,except, possibly, in a countable set.

Fix a point a ∈ ∂D where f is quasi-continuous, and an inner point O ∈ D.There exists a sequence γk, γk ⊂ D, k = 1, 2, . . ., of arcs with ends on Γseparating the point a from O and such that

limk→∞

osc (γk, f) = 0. (10.44)

Denote by Dk the subset of D separating γk from the point O. The inequal-ity (10.43) implies that

osc (Dk, f) ≤ osc (γk, f) + 2 arccosh (1 + length (γk)) (k = 1, 2, . . .).

Thus by (10.44) we obtain

limk→∞

osc (Dk, f) = 0 ,

and f is continuously extended to the point a ∈ Γ. Theorem is proved. 2

249

Open questions 10.45 1) Find a direct (not using auxiliary conformal map-pings of a surface to the plane of isothermal coordinates) proof of Theorem10.10. 2) Study behavior of generalized solutions near points of quasi-continuity and near jump points. In what cases, do limits of solutions alongnon-tangent ways exist? Along tangent ways? 3) Find multi-dimensionalversions of Fatou’s type theorem for minimal graphs.

250

Index

BL-mapping, 101BL-solution, 103BLk-mapping, 101BLk-solution, 103W 1,2-majorized function, 42, 125m-domain, 216q-quasi-conformal mapping, 44

abutting subdomain, 145Dirichlet integral, 149non-overlapping domains, 154, 157reduced module with respect to a

boundary point or a primeend, 152

abstract surface, 8, 13adjacent point of a body, 72admissible function for arc family,

14Ahlfors theorem, 143Ahlfors-Warschawski type theorem,

170analog of J.C.C. Nitsche theorem,

216area element, 8

Bernstein theorem, 197bi-Lipschitz mapping, 7body of a prime end, 70

canonical approximation, 45canonical homeomorphism, 46capacity of condenser, 24Caratheodory prime end, 70

Cauchy inequality with ε > 0, 53Cauchy-Riemann system with respect

to a metric, 168chain rule, 94change of variables, 55characteristics of a quasi-conformal

mapping, 44classification of the prime ends, 71closeness measure, 94complex dilatation, 103complex potential, 168condenser, 24condition of the parabolic type, 133condition of the type hyperbolicity,

130conformal mapping into surfaces, 63conformal mapping onto a surface,

47conformal type of a surface, 130, 233conjugate function, 167, 187, 211cross section of a surface, 71

Dirichlet integral, 55dislocation, 9distance, 8distortion coefficient, 212domains of arbitrary connectedness,

65dual function G, 11

elementary domain, 198elliptic for its solution, 166embedded, 32

251

252 Index

embedded surface, 32entire solution, 195equipotential uniform continuity, 61estimate of a summary jump, 230,

246estimate of measure distortion, 121extremal length of arc family, 14

Fatou theorem, 228Finn’s lemma, 249Finsler pseudometric, 12function collection H is coordinated

at a ∈ D, 8

gas dynamic equation, 165Gauss curvature of a surface, 134generalized derivation, 44generalized derivatives, 232generalized solution, 34, 166, 196geodesic circle, 80geodesic disc, 80geodesic ring, 80growth of a conjugate function, 213,

215

Hausdorff measure, 9height function, 36holomorphic function, 195holomorphic with respect to a met-

ric, 190homeomorphism of the class Q∗, 78hydrodynamic normalization, 133hyperbolic conformal type, 233

immersed surface, 32index of a critical point, 195inner conformal radius, 145inner metric, 13isothermal coordinates, 33

J.C.C. Nitsche problem, 194jump point, 229, 243

kernel convergence, 45

Laplace-Beltrami equation, 34Lavrentiev relative distance, 67Length and Area Principle, 28length element, 8level set, 28Lipschitz mapping, 7locally bi-Lipschitz surface, 64Lusin N -property, 46, 57

mappings of a class BL, 101maximal surface equation, 220Mazurkiewicz distance, 13Meeks conjecture, 220minimal surface, 36minimal surface type equation, 196,

227module of a condenser, 200module of an arc family in a metric,

169module of arc family, 13, 14module of condenser, 24monotonicity in the Lebesgue sense,

240

narrow domains, 221non-overlapping domains, 163nonparametric surface, 40

open Jordan arc in D, 145

parabolic conformal type, 234part integration formula, 52Phragmen-Lindelof type theorem, 178point of quasi-continuity, 229point of the quasi-continuity, 241prime end of the I-st type, 72prime end of the II-nd type, 72prime end of the III-rd type, 72prime end of the IV-th type, 72principal point of a body, 72problem of J.C.C. Nitsche, 228

253

properties of quasi-conformal maps,232

pseudoharmonic function, 222pseudometric, 10pseudometric space, 10

quasi-regular metric, 211

reduced module, 145restriction of W 1,2-majorized func-

tion, 125

Saint-Venant principle, 204Scherck surface, 228simple Jordan curve in D, 145simple point of a prime end, 72simplicity condition of a boundary

point, 72speed potential, 165stability in a closed domain, 113stability of conformal mappings, 93stability on compacts, 111structure conditions, 226structure restrictions for equations,

196subharmonic function, 34summary index of critical points, 223superharmonic function, 34symmetry principle, 16

theorem on narrow domains, 217types of the prime ends, 72

Warschawski theorem, 143

References

[1] D.R. Adams and L.I. Hedberg, Function Spaces and Potential Theory,Springer-Verlag, Berlin-Heidelberg-New York etc., 1996.

[2] L.V. Ahlfors, Untersuchungen zur Theorie der konformen Abbildungenund der ganzen Funktionen, Acta. Soc. Sci. Fenn. N. S. A I, 1, n. 9, 1930,1-40.

[3] L.V. Ahlfors, Lectures on quasiconformal mappings, D. Van NostrandComp. Inc., Princeton, 1966.

[4] L.V. Ahlfors, Conformal invariants, McGraw-Hill, New York, 1973.

[5] R.S. Akopyan, Stabilization conditions of a minimal surface over half-strip, in ”Scien. schools of Volgograd State univ. Geometric analysis andits applications”, izd-vo VolGU, Volgograd, 1999, 105-110 (in Russian).

[6] G. Alessandrini and V. Nessi, Univalent σ-harmonic mappings, Arch. Ra-tion. Mech. and Anal., v. 158, 2001, 155-171.

[7] G.S. Asanov, Finslerlike Geometry, izd-vo MGU, Moscow, 2004 (in Rus-sian).

[8] K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173,1994, 37—60.

[9] K. Astala and V. Nesi, Composites and quasiconformal mappings: new op-timal boundes, Preprint 223, Departm. of Math., University of Jyvaskyla,2000, 26 pp.

[10] F.G. Avkhadiev, Conformal maps and boundary problems, Kazan’ fund”Mathematics” , 1996, 216 pp.

[11] C. Bandle, Isoperimetric inequalities and applications, Pitman Adv. Pub.Program: Boston—London—Melbourne, 1980.

254

255

[12] E.F. Beckenbach, R. Bellman, Inequalities, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1961.

[13] P.P. Belinskii, On distortion under quasiconformal maps, Dokl. AN SSSR,v. 91, n. 5, 1953, 997-998 (in Russian).

[14] P.P. Belinskii, On area measure under quasiconformal mappings, Dokl.AN SSSR, v. 121, n. 1, 1958, 16-17 (in Russian).

[15] P.P. Belinskii, Solution of extremal problems of the quasiconformal mapstheory by the variational method, Sib. math. j., v. 1, n. 3, 1960, 303-330(in Russian).

[16] P.P. Belinskii, On the closeness order of a space quasiconformal map tothe conformal map, Sib. math. j., v. 14, n. 3, 1973, 475-483 (in Russian).

[17] P.P. Belinskii, General properties of quasiconformal maps, Nauka, Sib.otd., Novosibirsk, 1974 (in Russian).

[18] V.I. Belyi, V.M. Miklyukov, Some properties of conformal and quasicon-formal maps and direct theorems of the constructive function theory, Izv.AN SSSR, Ser. math., v. 38, n. 6, 1974, 1343-1361 (in Russian).

[19] S.N. Bernstein, On a geometrical theorem and its applications to partialdifferential equations, Sobr. soch., v. III, Izd-vo AN SSSR, 1960, 251-258(in Russian).

[20] L. Bers, Mathematical aspects of subsinic and transonic gas dinamics,New York · John Wiley & Sons, inc. London · Chapman & hall, Limited,1958.

[21] L. Bers, Uniformization by Beltrami equation, Comm. Pure Appl. Math.,v. 14, 1961, 215-228.

[22] A.A. Borisenko, The inner and exterior geometry of multi dimensionalsubmanifolds, Izd-vo Ekzamen, Moscow, 2003 (in Russian).

[23] C. Caratheodory, Untersuchungen uber die konformen Abbildungen vonfesten und veranderlichen Gebieten, Math. Ann., v. 72, 1912, 107-144.

[24] C. Caratheodory, Uber die Begrenzung einfach zusammenhangender Ge-biete, Math. Ann., v. 73, 1913, 323-370.

[25] C. Caratheodory, Conformal Representation, Cambridge at the UniversityPress, 1932.

256 References

[26] T. Carleman, Sur une inegalite differentielle dans la theorie des functionsanalytiques, C.R.Acad. Sci. Paris, v. 196, 1933, 995-997.

[27] S. Chandrasekhar, The mathematical theory of black holes, ClarendonPress Oxford Oxford University Press New York, 1983.

[28] E.F. Collingwood, A.J. Lohwater, The theory of cluster sets, CambrigeUniv. Press, Cambrige, 1966.

[29] R. Courant, Dirichlet’s principle, conformal mapping, and minimal sur-faces, Interscience, New York, 1950.

[30] G. David and P. Mattila, Removable sets for Lipschitz harmonic functionsin the plane, Revista Matem. Iberoamericana, v. 16, n. 1, 2000, 137-215.

[31] E.P. Dolgenko, On singularity removal of analytic functions, Uspehi math.nauk, v. 18, n. 4, 1963, 135-142 (in Russian).

[32] Yu.V. Dymchenko, Equality of capacity and module of a condenser ona surface, Zapiski nauchn. seminarov LOMI, v. 276, 2001, 112-133 (inRussian).

[33] J. Eells and B. Fuglede, Harmonic Maps between Riemannian Polyhe-dra, Cambridge Tracts in Mathematics 142, Cambridge Univ. Press, UK,2001.

[34] A. Eremenko and D.H. Hamilton, On the area distortion by quasiconfor-mal mappings, Proceedings of the American Mathematical Sosiety, 123,n. 9, 1995, 2793-2797.

[35] M.A. Evgrafov, Analytic functions, Nauka, Moscow, 1968 (in Russian).

[36] D. Faraco, Beltrami operators and microstructure, Academic dissertation,Depart. of Math., Faculty of Sci., University of Helsinki, Helsinki, 2002.

[37] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.

[38] R. Finn, Remarks relevant to minimal surfaces, and to surfaces of pre-scribed mean curvature, J. Analyze Math., v. 14, 1965, 139-160.

[39] A.T. Fomenko, Variational methods in the topology, Nauka, Moscow, 1982(in Russian).

[40] B. Fuglede, Extremal length and functional completetion, Acta Math.,v. 98, n.3-4, 1957, 171-219.

[41] F.R. Gantmacher, Theory of matrices, Nauka, Moscow, 1967 (in Russian).

257

[42] F.W. Gehring, O. Lehto, On the total differentiability of functions of acomplex variable, Ann. Acad. Sci. Fenn. A I, 272, 1959, 1-9.

[43] F. Gehring and E. Reich, Area distortion under quasiconformal mappings,Ann. Acad. Sci. Fenn. Ser. AI 388, 1966, 1-14.

[44] E. Giusti, Minimal surfaces and functions of bounded variation,Birkhauser, Boston-Basel-Stuttgart, 1984.

[45] S.K. Godunov, E.I. Romenskii, Elements of mechanics and conservationlaws, Izd-vo ”Nauchnaya kniga”, Novosibirsk, 1998 (in Russian).

[46] V.M. Goldstein, Yu.G. Reshetnyak, Introduction to theory of functionswith generalized derivatives and quasiconformal maps, Nauka, Moscow,1983 (in Russian).

[47] G.M. Golusin, Geometric theory of complex variable functions, Nauka,Moscow, 1966 (in Russian).

[48] C. Grammatico, A result on strong unique continuation for the Laplaceoperator, Commun. in partial defferential equations, v. 22 (9&10), 1997,1475-1491.

[49] W. Gresky, Konforme Abbildungen der Oberflache Eines RektangularenHexaeders auf die Kugeloberflache, Inaugural-Dissertatin zur Erlangungdef Doktorwurde der Hohen Philosophischen Fakultat der UniversitatLeipzig, Weida i. Thur, 1928, 1-74.

[50] E.G. Grigoryeva, A.A. Klyachin, and V.M. Miklyukov, Problem of Func-tional Extension and Space-Like Surfaces in Minkowski Space, Journalsof Analysis and its Applications, v. 21, n. 3, 2002, 719-752.

[51] A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer.Math. Soc., v. 36, n. 2, 1999, 135-249.

[52] M. Gromov, Metric Structures for Riemannian and Non-RiemannianSpaces, Boston-Basel-Berlin, Birkhauser, 1999.

[53] I.M. Grudskii, Structure of inner coordinates on combined Rieman-nian surfaces, In ”Differential, integral equations and complex analysis”,Elista, 1986, 30-45 (in Russian).

[54] I.M. Grudskii, Christoffel-Schwarz formula for polyhedral surfaces, Dokl.AN SSSR, v. 307, n. 1, 1989, 15-17 (in Russian).

258 References

[55] V. Gutlyanskii, O. Martio, T. Sugawa and M. Vuorinen, On the De-generate Beltrami Equation, Reports of the Department of Mathematics,University of Helsinki, Preprint 282, 2001, 32 pp.

[56] P. Hajlasz, Sobolev Mappings, Co-Area Formula and Related Topics, Pro-ceedings on analysis and geometry, Sobolev Institute Press, Novosibirsk,2000, 227-254.

[57] H. Haken, Advanced Synergetics, Springer - Verlag, Berlin Heidelberg NewYork Tokyo, 1983.

[58] S.Ya. Havinson, On erasure of singularities, Litovskii math. sb., III, n. 1,1963, 271-287 (in Russian).

[59] J. Heinonen, T. Kilpelainen, and O. Martio, Nonlinear potential theory ofdegenerate elliptic equations, Clarendon Press, Oxford etc., 1993.

[60] J. Heinonen, P. Koskela, Sobolev Mappings with Integrable Dilatations,Arch. Rational Mach. Anal., v. 125, 1993, 81-97.

[61] I.O. Herzog, Phragmen-Lidelof theorems for second order quasilinear el-liptic partial differential equations, Proc. Amer. Math. Soc., v. 15, 1964,721-728.

[62] D.P. Il’yutko, Locally minimal nets in N -normed spaces, Math. zametki,v. 74, n. 5, 2003, 656-668 (in Russian).

[63] O.Yu. Imanuvilov, On Carleman estimates for hyperbolic equations,Asymptotic Analysis, v. 32, 2002, 185-220.

[64] T. Iwaniec, G. Martin, Geometric Function Theory and Nonlinear Analy-sis, Oxford Mathematical Monographs, Oxford University Press, Oxford,2001.

[65] O.V. Ivanov, G.D. Suvorov, Full lattice of conformally invariant compact-ifications of a domain, Naukova Dumka, Kiew, 1982 (in Russian).

[66] J.A. Jenkins, Univalent functions and conformal mapping , Springer-Verlag, Berlin-Gottingen-Heidelberg, 1958.

[67] A.P. Karmazin, Quasiisometries, theory of preends and metric structuresof space domains, Izd-vo Surgutskogo un-ta, Surgut, 2003 (in Russian).

[68] R. Kersner, A. Shishkov, Instantaneous Shrinking of the Support of En-ergy Solutions, J. of Math. Anal. and Appl., v. 198, 1996, 729-750.

259

[69] V.A. Klyachin, V.M. Miklyukov, Tubes and bands in the space-time, Izd-vo Volgogradsk. univ., Volgograd, 2004 (in Russian).

[70] A.P. Kopylov, Stability of Classes of Mappings and Holder Continuity ofHigher Derivatives of Elliptic Solutions to Systems of Nonlinear Differ-ential Equations, Sib. Math. J., v. 43, n. 1, 2002, 68-82 (in Russian).

[71] P. Koskela, J. Maly, Mappings of finite distortion: The zero set of theJacobian, Journal of the European Mathematical Society, v. 5, n. 2, 2003,95-105.

[72] V.I. Kruglikov, On existence and uniqueness of maps quasiconformal inmean, in ’Metr. voprosy theor. func. i otobr.’, n. IV, Naukova Dumka,Kiew, 1973, 123-147 (in Russian).

[73] V.I. Kruglikov, Condenser capacities and space maps quasiconformal inmean, Math. sb., v. 130 (172), n. 2, 1986, 185-206 (in Russian).

[74] V.I. Kruglikov, V.M. Miklyukov, Stability theorems for maps of the classBL, in ’Metr. voprosy theor. functii i otobr.’, n. III, Izd-vo NaukovaDumka, Kiew, 1971, 55-70 (in Russian).

[75] V.I. Kruglikov, V.M. Miklyukov, On some classes of plane topologicalmaps with generalized derivatives, in ’Metr. voprosy theor. functii i otobr.’,n. IV, Izd-vo Naukova Dumka, Kiew, 1973, 102-122 (in Russian).

[76] S.L. Krushkal’, On mappings quasiconformal in mean, Sib. math. j., v. 8,n. 4, 1967, 798-806 (in Russian).

[77] S.L. Krushkal’, Quasiconformal mappings and Riemann surfaces, JohnWiley&Sons, New York-Toronto-London-Sydney, 1979.

[78] N. Kuiper, On C1-isometric embeddings, 1, 2, Proc. Koninkl. Nederl.Akad. Wetensch. Ser. A, v. 58, n. 4, 5, 1955, 545-556, 683-689.

[79] B.P. Kufarev, Potentials and boundary correspondence, Izv. AN SSSR,Ser. math., v. 41, n. 2, 1977, 438-461 (in Russian).

[80] R. Kuhnau, Trianguliere Riemannsche Mannigfaltigkeiten mit ganz-linearen Bezugssub-Stitutionen and st”uchweise konstanter komplexer Di-latation, Math. Nachrichten, B. 46, Heft 1-6, 1970, 243-261.

[81] R. Kuhnau, Uber zwei klassen schlichter konformer Abbildungen, Math.Nachr., B. 49, N. 1-6, 1971, 173-185.

[82] K. Kuratowski, Topology, v. I, Academic press, Warszawa, 1966.

260 References

[83] K. Lancaster, Nonparametric minimal surfaces in R3 whose boundarieshave a jump discontinuity, Internat. J. Math. Math. Sci., v. 11, n. 4, 1988,651-656.

[84] M. Lavrentieff, Sur une classe de representations continue, Math. sb.,v. 42, n. 4, 1935, 407-424.

[85] M.A. Lavrentiev, On continuity of univalence functions in closed domains,Dokl. AN SSSR, v. 4, 1936, 207-210 (in Russian).

[86] M.A. Lavrentiev, B.V. Shabat, Problems of hydrodinamics and theirmathematical models, Nauka, Moscow, 1973 (in Russian).

[87] J. Lawrynowicz, On a class of quasiconformal mappings with invariantboundary points, II, Applications and generalizations, Ann. polon. math.,v. 21, n. 3, 1969.

[88] N.A. Lebedev, Area principle in the theory of univalent functions, Nauka,Moscow, 1975 (in Russian).

[89] H. Lebesgue, Sur le probleme de Dirichlet, Rend. circ. mat. Palermo, 24,1907, 371-402.

[90] O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane, Sec-ond Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

[91] J. Lelong-Ferrand, Representation conforme et transformations aintegrale de Dirichlet bornee, Gauthier-Villars, Paris, 1955.

[92] B.E. Levickii, V.M. Miklyukov, Reduced Modulus on a Surface, VestnikTomsk. Gosud. Univ., n. 301, Tomsk, 2007, 87-91 (in Russian).

[93] B.E. Levickii, I.P. Mityuk, ’Narrow’ theorems on space module, Dokl. ANSSSR, v. 248, n. 4, 1979, 780-783 (in Russian).

[94] V.P. Luferenko, G.D. Suvorov, On concept of a prime end body in theCaratheodory theory, in ’Metr. voprosy theor. func. i otobr.’, ’Metr. vopr.teor. func. i otobr.’, n. III, Naukova dumka, Kiew, 1971, 71-79 (in Rus-sian).

[95] Y. Li and L. Nirenberg, The Distance Function to the Boundary, FinslerGeometry, and the Singular Set of Viscosity Solutions of Some Hamilton-Jacobi Equations, Communications of Pure and Applied Mathematics,v. LVIII, 2005, 85-146.

[96] P. Li and J. Wang, Finiteness of disjoint minimal graphs, Math. Res.Letters, v. 8, 2001, 771-777.

261

[97] J. Maly, Sufficient Conditions for Change of Variables in Integral, in’Proc. on Analysis and geometry’, Izd-vo Inst. of math., Novosibirsk, 2000,370-386.

[98] O. Martio, V. Miklyukov, On existence and uniqueness of degenerate Bel-trami equation, Complex Variables, v. 49, 2004, 647-656.

[99] O. Martio, V.M. Miklyukov, and M. Vuorinen, Critical points ofA−solutions of quasilinear elliptic equations, Houston Journal of Mathe-matics, v. 25, n. 3, 1999, 583-601.

[100] O. Martio, V.M. Miklyukov, and M. Vuorinen, Relative distance andboundary properties of nonparametric surfaces with finite area, Journalof Mathematical Analysis and Applications, v. 286, n. 2, 2003, 524-539.

[101] O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, BMO-quasiconformal mappings and Q-homeomorphisms in space, Prep. 288,Dept. Math. of Helsinki Univ., 2001, 24 pp.

[102] O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, On the boundarybehavior of Q-homeomorphisms, Prep. 318, Dept. Math. of Helsinki Univ.,2002, 12 pp.

[103] V.G. Maz’ya, On regularisity on the boundary of solutions of elliptic equa-tions and conformal maps, Dokl. AN SSSR, v. 152, n. 6, 1963, 1297-1300(in Russian).

[104] E.J. McShane, Parametrizations of saddle surfaces, with applications tothe problem of Plateau, Trans. Amer. Math. Soc., 35, 1933, 716-733.

[105] D.E. Men’shov, Les conditions de monogeneite, Act. sci. et ind., 1936,v. 329, 1-52.

[106] V.M. Miklyukov, On ε-quasiconformal mappings on a ball onto a ball, in’Metr. voprosy theor. func. i otobr.’, n. I, Naukova Dumka, Kiew, 1969,140-162 (in Russian).

[107] V.M. Miklyukov, Some estimates of a conformal mapping of a domain toa strip, Dokl. AN SSSR, v. 223, n. 3, 1975, 295-297 (in Russian).

[108] V.M. Miklyukov, On some boundary problems of the conformal mappingtheory, Sib. math. j., v. 18, n. 5, 1977, 1111-1124 (in Russian).

[109] V.M. Miklyukov, Two theorems on boundary properties of a minimal sur-face in the nonparametric form, Math. zametki, v. 21, n. 4, 1977, 551-556(in Russian).

262 References

[110] V.M. Miklyukov, On the conformal type of surfaces, Liouville’s theoremand Bernstein’s theorem, Sov. Math. Dokl., v. 19, n. 5, 1978, 1150-1153.

[111] V.M. Miklyukov, On a new approach to Bernstein’s theorem and relatedquestions for equations of minimal surface type, Math. USSR Sb., v. 36,n. 2, 1980, 251-271.

[112] V.M. Miklyukov, On the asymptotic properties of subsolutions of quasi-linear equations of elliptic type and mappings with bounded distortion,Math. USSR Sb., v. 39, 1981, 37-60.

[113] V.M. Miklyukov, Some singularities in the behavior of solutions of equa-tions of minimal-surface type in unbounded domains, Math. USSR sb.,v. 44, n. 1, 1983, 61-73.

[114] V.M. Miklyukov, Some questions of the qualitative theory of minimal sur-face type equations, in ’Boundary problems of the mathematical physics’,Naukova Dumka, Kiew, 1983 (in Russian).

[115] V.M. Miklyukov, Remarks on the lenght and area principle, in theG.D. Suvorov book ’Generalized lenght and area principle in the theoryof maps’, Naukova Dumka, Kiew, 1985, 252-260 (in Russian).

[116] V.M. Miklyukov, On critical points of solutions of maximal surface typeequations in the Minkowski space, Teoriya otobr. i pribl. func., NaukovaDumka, Kiew, 1989, 112-125 (in Russian).

[117] V.M. Miklyukov, Stagnation zones of solutions to the Laplace-Beltramiequation in long strips, Siberian Advances in Mathematics, v. 12. n. 4.2002. 1-17.

[118] V.M. Miklyukov, Isothermic coordinates on singular surfaces, Sbornik:Mathematics, v. 195, n. 1, 2004, 65-84.

[119] V.M. Miklyukov, Relative Lavrent’eff Distance and Prime Ends on Non-parametric Surfaces, Ukrainian Mathematical Bulletin, v. 1, n. 3, 2004,353-376.

[120] V.M. Miklyukov, On some applications of the M.A. Lavrentiev relativedistance, Dokl. RAN, v. 402, n. 4, 2005, 448-451 (in Russian).

[121] V.M. Miklyukov, On a property of the height function of multi dimen-sional Plateau problem, Math. i prikladn. anal., n. 2, Izd-vo Tyumenskogouniv., Tyumen’, 2005, 110-119 (in Russian).

263

[122] V.M. Miklyukov, Area distortion under conformal maps of surfaces,Trudy seminara po vectorn. i tenzorn. anal., n. XXVI, Izd-vo MGU,Moscow, 2005, 238-249 (in Russian).

[123] V.M. Miklyukov, Speed of Approximation to Degenerate QuasiconformalMappings and Stability Problems, Complex Analysis and Potential The-ory, Proceedings of the Conference Satellite to ICM 2006, World ScientificPublishing Co. Pte. Ltd., 2007, 33-45.

[124] V.M. Miklyukov, S.S. Polupanov, R.A. Tarapata, On stabilization of gasdinamics equation solutions, Dokl. RAN, v. 388, n. 5, 2003, 596-598 (inRussian).

[125] V.M. Miklyukov, G.D. Suvorov, On existence and uniqueness of quasicon-formal maps with unbounded characteristics, in ’Issled. po sovrem. teoriifunct. and its appl.’, Naukova Dumka, Kiew, 1972, 45-53 (in Russian).

[126] J. Milnor, On deciding whether a surface is parabolic or hyperbolic, Amer.Math. Monthly, v. 84, n. 1, 1977, 43-46.

[127] Ch.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, W.H.Freemanand Company, San Francisco, 1973.

[128] I.P. Mityuk, Generalized reduced module and some of its applications, Izv.vusov, Mathem., n. 2, 1964, 110-119 (in Russian).

[129] I.P. Mityuk, On quasiconformal maps in the space, Dopov. AN USSR,n. 8, 1964, 563-566 (in Ukrainian).

[130] V.N. Monachov, Boundary problems with free boundaries for elliptic sys-tems equations, Nauka, Novosibirsk, 1977 (in Russian).

[131] M. Morse, Topological methods in the theory of functions of a complexvariables, Princeton Univ. Press, Princeton, 1947.

[132] F. Morgan, Geometric Measure Theory. A Beginners’s Guide, SecondEdition, San Diego-Boston ets, Academic Press, 1995.

[133] C.B. Morrey, On the solutions of quasi-linear elliptic partial differentialequations, Trans. Amer. Math. Soc., v. 43, 1938, 126-186.

[134] S. Muller, V. Sverak, On surfaces of finite total curvature, J. DifferentialGeometry, v. 42, 1995, 229-258.

[135] J. Nash, C1-isometric imbeddings, Ann. Math., v. 60, 1954, 383-396.

264 References

[136] S.R. Nasyrov, The metric space of Rimann surfaces over the sphere, Rus-sian Acad. Sci. Sb. Math., v. 82, n. 2, 1995, 337-356.

[137] J.C.C. Nitsche, On new results in the theory of minimal surfaces, Bull.Amer. Math. Soc., v. 71, n. 2, 1965, 195-270.

[138] O.A. Oleinik, G.A. Iosif’yan, Sent-Venan principle analog and uniquenessof solutions of boundary problems in unbounded domains for parabolicequations, Uspehi math. nauk, v. 31, n. 6, 1976, 142-166 (in Russian).

[139] O.A. Oleinik, G.A. Iosif’yan, Boundary value problems for second orderelliptic equations in unbounded domains and Saint-Venant’s Principle,Ann. Suola Norm. Sup. Pisa Cl. Sci. (4), n. 2, 1977, 269-290.

[140] O.A. Oleinik, E.V. Radkevich, Analyticity and Liouville and Phragmen-Lindelof type theorems for general elliptic system equations, Math. sb.,v. 95 (137), n. 1, 1974, 130-145 (in Russian).

[141] R. Osserman, Minimal surfaces, Uspehi Math. Nauk, v. 22, n. 4, 1967,55-135 (in Russian).

[142] Y. Pan and T. Wolf, A Remark on Unique Continuation, The J. of Geo-metric Analysis, v. 8, n. 4, 1998, 599-604.

[143] V.L. Panteleev, Theory of Earth Figure, Izd-vo MGU, Moscow, 2000,160 pp. (in Russian)

[144] V.I. Pelich, Phragmen-Lindelof theorems on minimal surfaces, in ’Ge-ometr. anal. i ego prilog.’, Izd-vo Volgograd. univ., volgograd, 1999, 352-368 (in Russian).

[145] G. Piranian, The distribution of prime ends, Mich. Math. J., v. 7, 1960,83-95.

[146] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin-Heidelberg-New York, 1992.

[147] I.I. Privalov, Boundary properties of analytic functions, GITTL, Moscov,1950 (in Russian).

[148] Yu.G. Reshetnyak, Stability theorems in geometry and analysis, Nauka,Novosibirsk, 1982 (in Russian).

[149] Yu.G. Reshetnyak, Two-dimensional manifolds of Bounded curvature,in ’Geometry IV’, Non-regular Riemannian Geometry, Springer-Verlag,Berlin-Heidelberg ets, 1993, 6-163.

265

[150] R.T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton,1970.

[151] H. Rund, The differential geometry of Finsler spaces, Springer-Verlag,Berlin-Gottingen ets, 1959.

[152] C.A. Rogers, Hausdorff measures, Cambridge at the University Press,1970.

[153] B. Rodin, S.E. Warschawski, Extremal length and the boundary behaviorof conformal mapping, Ann. Acad. Sci. Fenn. Ser. AI Math, v. 2, 1976,467-500.

[154] B. Rodin, S.E. Warschawski, Extremal length and univalent functions II.Integral estimates of strip mappings, J. Math. Soc. Japan, v. 31, 1979,87-99.

[155] B. Rodin, S.E. Warschawski, Estimates for conformal maps of strip do-mains without boundary regularity, Proc. London Math. Soc., v. 39, 1979,356-384.

[156] B. Rodin, S.E. Warschawski, On the derivative of the Riemann mappingsfunction near a boundary point and the Visser-Ostrowski problem, Math.Ann., v. 248, 1980, 125-137.

[157] B. Rodin, S.E. Warschawski, A necessary and sufficient condition for theasymptotic version of Ahlfors’ distortion property, Trans. Amer. Math.Soc., v. 276, 1983, 281-288.

[158] B. Rodin, S.E. Warschawski, Conformal mapping and locally asymptoti-cally conformal curves, Proc. London Math. Soc., v. 49, 1984, 255-273.

[159] V.I. Ryazanov, On compactness and metrizability of kernel spaces of man-ifolds, Ukrainian Math. J., v. 50, n. 6, 1999, 944-951 (in Russian).

[160] V. Ryazanov, U. Srebro, E. Yakubov, Degenerate Beltrami equation andradial Q-homeomorphisms, Preprint 369, August 2003, Department ofMathematics, University of Helsinki.

[161] S. Saks, Theory of the integral, Warsaw, 1939.

[162] L. Sario, M. Nakai, C. Wang, L.O. Chung, Classification Theory of Rie-mannian Manifolds, Lectures Notes Math., 1977, 605 pp.

[163] M. Schiffer and D.C. Spencer, Functionals of finite Riemann surfaces,Princeton University Press, Princeton, New Jersey, 1954.

266 References

[164] L. Schwartz, Analyse mathematique I, Hermann, 1967.

[165] H.A. Schwarz, Conforme Abbildung der Oberflache eines Tetraeders aufdie Oberflache einer Kugel. - J reine angew. Math., v. 70, 1869, 121-136.

[166] B.V. Shabat, Introduction to the complex analysis, Nauka, Moscow, 1969(in Russian).

[167] E.V. Shikin, On parabolicity of immersiability and hyperbolicity of non-immersiability twodimensional manifolds of the negative curvature, Vestn.Moscow un-ta. Mathem. Mech., n. 5, 1990, 42-45 (in Russian).

[168] A.E. Shishkov, Behaviour of generalized solutions of the Dirichlet prob-lem for general quasilinear divergence elliptic equations at the boundaryneighborhood, Different. equat., v. 23, n. 2, 1987, 308-320 (in Russian).

[169] J. Serrin, M. Tang, Uniqueness of Ground States for Quasilinear EllipticEquation, Indiana Univ. Math. J., v. 49, n. 3, 2000, 897-923.

[170] L. Simon, Equation of mean curvature in 2 independent variables, PacificJ. Math., v. 69, n. 1, 1977, 245-268.

[171] V.I. Smirnov, Kurs vysshei matematiki, v. 5, Fismatgiz, Moscow, 1959 (inRussian).

[172] S.L. Sobolev, Some applications of the functional analysis in the mathe-matical physics, Izd-vo Leningrad. univ., Leningrad, 1950 (in Russian).

[173] J. Spruck, Two dimensional graphs over unbounded domains, J. Inst.Math. Jussien, v. 1, 2002, 631-640.

[174] S. Stoilow, Le cons sur les principes topologiques de la theorie de fonctionsanalytiques, Cauthier - Villars, Paris, 1938.

[175] Yu.F. Strugov, Quaiconformal in mean maps and extremal problems,Parth 1, Omsk, 1994, 154 pp. - Dep. in VINITI 05.12.94, No 2786-94;Parth 2, Omsk, 1994, 114 pp. - Dep. in VINITI 05.12.94, No 2787-94 (inRussian).

[176] G.D. Suvorov, Remarks to a M.A. Lavrentiev theorem, Uch. zap. Tomsk.univ., v. 25, 1956, 3-8 (in Russian).

[177] G.D. Suvorov, Families of plane topological maps, Izd-vo SO AN SSSR,Novosibirsk, 1965 (in Russian).

[178] G.D. Suvorov, Uniform stability of conformal maps of closed domans,Ukrain. math. j., v. 20, n. 1, 1968, 78-84 (in Russian).

267

[179] G.D. Suvorov, Generalized ’length and area’ principle in the theory ofmaps, Naukova Dumka, Kiew, 1985 (in Russian).

[180] G.D. Suvorov, Prime ends and sequences of plane domains of plane maps,Naukova Dumka, Kiew, 1986 (in Russian).

[181] A.V. Sychev, Module and space quasiconformal maps, Nauka, Novosi-birsk, 1983 (in Russian).

[182] A.F. Tedeev, Qualitative properties of solutions of the Neyman problemfor a quasilinear parabolic equation of the height order, Ukr. math. j.,v. 45, n. 11, 1993, 1571-1579 (in Russian).

[183] O. Teichmuller, Untersuchungen uber konforme und quasikonforme Ab-bildung, Dtsch. Math., n. 3, 1938, 621-678.

[184] V.G. Tkachev, A.N. Ushakov, Fuglede theorem in Finsler spaces, in ’Thepotential theory’, Naukova Dumka, Kiew, 1991, p. 15 (in Russian).

[185] T. Toro, Surfaces with generalized second fundamental form in L2 areLipschitz manifolds, J. Differential Geometry, v. 39, 1994, 65-101.

[186] Yu.Yu. Trohimchyk, Continuous maps and conditions of monogeneity,GFML, Moscow, 1963 (in Russian).

[187] D.C. Ullrich, Removable sets for harmonic functions, Mich. Math. J.,1991, v. 38, n. 3, 467-473.

[188] N.X. Uy, Removable set for Lipschitz harmonic functions, Mich. Math.J., v. 37, 1990, 45-51.

[189] I.N. Vekua, Generalized analytic functions, GIFML, Moscow, 1959 (inRussian).

[190] G.M. Vergbickii, V.G. Maz’ya, Asymptotic behaviour of second order el-liptic equations close to boundary, I, Sib. math. j., v. 12, n. 6, 1971,1217-1249.

[191] G.M. Vergbickii, V.G. Maz’ya, Asymptotic behaviour of second order el-liptic equations close to boundary, II, Sib. math. j., v. 13, n. 6, 1972,1239-1271.

[192] L.I. Volkovyskii, Quasiconformal mappings, Izd-vo L’vov univ., L’vov,1954 (in Russian).

[193] I.A. Volynec, On distortion under mappings of the class BL, Sib. math.j., v. 18, n. 6, 1977, 1259-1270 (in Russian).

268 References

[194] A.G. Vorob’ev, V.M. Miklyukov, On some properties of subsolutions ofminimal surface type equations , Sib. math. j., v. 23, n. 1, 1982, 25-31.

[195] S.E. Warschawski, Uber das Randverhalten der Ableitung der Abbildungs-function bei konformer Abbildung, Math. Z., v. 35, 1932, 321-456.

[196] S.E. Warschawski, On conformal mapping of infinite strips, Trans. Amer.Math. Soc., v. 51, 1942, 280-335.

[197] S.E. Warschawski, On the boundary behaviour of conformal mappings,Nagoya Math. J., v. 30, 1967, 83-100.

[198] A. Weitsman, On the growth of minimal graphs, Indiana Univ. Math. J.,v. 54(2), 2005, 617-625.

[199] H. Whitney, Analytic extensions of differentiable functions defined inclosed sets, Trans. Amer. Math. Soc., v. 36, 1934, 63-89.

[200] T.H. Wolff, Note on counterexamples in strong unique continuation prob-lems, Proc. of the Amer. Math. Soc., v. 114, n. 2, 1992, 351-356.

[201] T.H. Wolff, A counterexample in a unique continuation problem, Comm.in Analysis and Geometry, v. 2, n. 1, 1994, 79-102.

[202] V.A. Zhukov, On a proof of the theorem on differential existence for W 1p -

functions, Trudy Tomsk. un-ta, Ser. mech.-math., v. 189, 1966, 13-17 (inRussian).